Search problems

8 problems match “algebraic identities”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. Sum of squares from means ●○○○○ 10 ptsReal numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$.
2. Coefficients of a polynomial identity ●○○○○ 10 ptsIf $(x+a)(x+8)=x^{2}+bx+24$ for all values of $x$, find the value of $a+b$.
3. Coefficients of an algebraic identity ●○○○○ 10 ptsThe equation $ax+18=2(x-b)$ holds for all values of $x$. Work out the value of $a-b$.
4. Sum of squares after two increments ●●○○○ 20 ptsEach of 100 given numbers was increased by $1$. Then each number was increased by $1$ once more. Given that after the first increase the sum of the squares of the numbers did not change, find by how much the sum of the squares changed after the second increase.
5. Sum of coefficients in a shifted cubic expansion ●●○○○ 20 ptsIf the equation $1+x+x^{2}+x^{3}=a_{0}+a_{1}(1-x)+a_{2}(1-x)^{2}+a_{3}(1-x)^{3}$ holds for all real $x$, where $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$ are real constants, find the value of $a_{1}+a_{2}+a_{3}$.
6. Quadratic expression in shifted variables ●●○○○ 20 ptsGiven $a=2009x+2007$, $b=2009x+2008$, $c=2009x+2009$, find the value of $a^{2}+b^{2}+c^{2}-ab-bc-ca$.
7. Completing the square for a quartic ●●○○○ 20 ptsRewrite the expression $g^4 + 12g^2 + 9$ in the form $c(g^2 + p)^2 + q$. What is $q$?
8. Possible values of c in a polynomial identity ●●●○○ 50 ptsGiven integers $a$, $b$ and $c$ such that the equation $(x+a)(x+b)+c(x-10)=(x-11)(x+1)$ holds for all $x$, find all possible values of $c$.